Entanglement scaling behaviors of free fermions on hyperbolic lattices

Abstract

Recently, tight-binding models on hyperbolic lattices (discretized AdS space) have gained significant attention, leading to hyperbolic band theory and non-Abelian Bloch states. In this paper, we investigate these quantum systems from the perspective of quantum information, focusing particularly on the scaling of entanglement entropy (EE) that has been regarded as a powerful quantum-information probe into exotic phases of matter. It is known that on $d$-dimensional translation-invariant Euclidean lattice, the EE of band insulators scales as an area law ($\sim L^{d-1}$; $L$ is the linear size of the boundary between two subsystems). Meanwhile, the EE of metals (with finite Density-of-State, i.e., DOS) scales as the renowned Gioev-Klich-Widom scaling law ($\sim L^{d-1}\log L$). The appearance of logarithmic divergence, as well as the analytic form of the coefficient $c$ is mathematically controlled by the Widom conjecture of asymptotic behavior of Toeplitz matrices and can be physically understood via the Swingle's argument. However, the hyperbolic lattice, which generalizes translational symmetry, results in inapplicability of these analytic approaches and the potential non-trivial behavior of EE. Here we make an initial attempt through numerical simulation. Remarkably, we find that both cases adhere to the area law, indicating the effect of background hyperbolic geometry that influences quantum entanglement. To achieve the results, we first apply the vertex inflation method to generate hyperbolic lattice on the Poincaré disk, and then apply the Haydock recursion method to compute DOS. Finally, we study the scaling of EE for different bipartitions via exact diagonalization and perform finite-size scaling. We also investigate how the coefficient of the area law is correlated to bulk gap in gapped case and to the DOS in gapless case respectively. Future directions are discussed.

18 pages. Accepted by Physical Review Research (2025). This paper is part of a series of work on quantum entanglement of fermions with 2311.01199, 2408.11652, 2312.08632, 2311.01997, 2205.01654, 2112.13411, 2009.00546, 1410.8670, and 1403.1039

References in corpus (46)

• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Entanglement entropy of fermions in any dimension and the Widom conjecture
• Entanglement Phase Transition Induced by the Non-Hermitian Skin Effect
• Entanglement scaling in critical two-dimensional fermionic and bosonic systems
• Simulating hyperbolic space on a circuit board
• Crystallography of Hyperbolic Lattices
• Observation of novel topological states in hyperbolic lattices
• Volume-to-Area Law Entanglement Transition in a non-Hermitian Free Fermionic Chain
• Automorphic Bloch theorems for hyperbolic lattices
• Entanglement Entropy Scaling Laws and Eigenstate Typicality in Free Fermion Systems
• Hyperbolic Topological Band Insulators
• Chern insulator in a hyperbolic lattice
• Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices
• Free-fermion Entanglement Spectrum through Wannier Interpolation
• Rényi entropies and negative central charges in non-Hermitian quantum systems
• Higher-order topological insulators in hyperbolic lattices
• Holography on tessellations of hyperbolic space
• Emergent entanglement phase transitions in non-Hermitian Aubry-André-Harper chains
• Higher-order Topological Hyperbolic Lattices
• Quantum Entanglement of Non-Hermitian Quasicrystals
• Non-Abelian hyperbolic band theory from supercells
• Quantum phase transitions of interacting bosons on hyperbolic lattices
• Symmetry and topology of hyperbolic Haldane models
• Entanglement Entropy of Non-Hermitian Free Fermions
• Density of states of tight-binding models in the hyperbolic plane
• Anderson localization transition in disordered hyperbolic lattices
• Hyperbolic non-Abelian semimetal
• Simulating Holographic Conformal Field Theories on Hyperbolic Lattices
• Measuring entanglement entropy and its topological signature for phononic systems
• Entanglement phase transitions in non-Hermitian quasicrystals
• Subarea law of entanglement in nodal fermionic systems
• Position momentum Duality in the Entanglement Spectrum of Free Fermions
• Converging Periodic Boundary Conditions and Detection of Topological Gaps on Regular Hyperbolic Tessellations
• Polarization and entanglement spectrum in non-hermitian systems
• Hyperbolic Lattice for Scalar Field Theory in AdS${}_3$
• Topological linear response of hyperbolic Chern insulators
• Hyperbolic Fractional Chern insulators
• Entanglement phase transitions in non-Hermitian Floquet systems
• Dynamic mass generation on two-dimensional electronic hyperbolic lattices
• Detecting bulk and edge exceptional points in non-Hermitian systems through generalized Petermann factors
• Thermodynamics and entanglement entropy of the non-Hermitian SSH model
• Hyperbolic lattices and two-dimensional Yang-Mills theory
• Entanglement Signature of Hinge Arcs, Fermi Arcs, and Crystalline Symmetry Protection in Higher-Order Weyl Semimetals
• Entangelment Entropy on Generalized Brillouin Zone
• Holographic analysis of boundary correlation functions for the hyperbolic-lattice Ising model
• Entanglement Fractalization